# Pulse excitation of damped harmonic oscillator: delayed maximal response

When a damped harmonic oscillator (HO) is excited by a pulse (Gaussian*sinusoid), the maximum of the oscillating response is delayed. This make sense because the impulse response $h(t)$ of the HO is asymmetric with respect to $t=0$, and the system needs time to build up a response: $$h(t)=\Theta(t)\frac{\sin(\omega_0 t)}{\omega_0}e^{-t/\tau}$$ with $\Theta(t)$ Heaviside's step function, $\tau$ the lifetime and $\omega_0$ the real part of the eigenfrequency.

I remarked that, as the central frequency $\omega$ of the pulse becomes closer to the resonant frequency $\omega_0$ and as the pulse becomes longer (width of the pulse $\Delta t$), this delay* tends to the lifetime $\tau$ of the HO:

Since $\tau$ is one of the only two characteristic of the HO, it makes sense that it appears in the expression of the delay. Graphically it is apparent that when the impulse response is shifted w.r.t. the center of the pulse by an amount too small or large compared to the lifetime, the overlap between the pulse and the impulse response is minimal.

Research in books/ internet concerning this effect lead to nothing. Nevertheless, since the damped harmonic oscillator-like systems appear in a lot of domains of physics (mechanics, electrical engineering, optics...), and is quite fundamental, there must be an explanation somewhere, I may just lack the right keyword.

Question: Is there a (simple) explanation/ proof of this phenomenon (in addition to what I said above) ? Is there a reason why this delay cannot exceed the lifetime ? (All of that assuming that I didn't messed up my computations). The best would be a reference.

*I took the delay at the maximum of the envelope of the response, the envelope being obtained by interpolation of the maxima.

• I do not understand this sentence: "Graphically it is apparent that when the impulse response is shifted w.r.t. the center of the pulse by an amount too small or large compared to the lifetime". Where are you shifting the impulse response? – Crimson Apr 4 '18 at 12:39
• Could you be more specific what you mean when you ask for an expanation for "this phenomenon"? – Crimson Apr 4 '18 at 12:40
• When computing the convolution: the response a a time $t$ is given by shifting the (reversed) $h(t)$ by $t$ and integrating the product of $h(t)$ and the exciting pulse. – David Apr 4 '18 at 12:41
• I would like an analytical "proof" that this delay cannot exceed, and tends to the lifetime. The best would be reference that I could cite – David Apr 4 '18 at 12:42
• When the "impulse" $\Delta t$ is much greater than the lifetime $\tau$ of the damped oscillator, it is intuitively obvious that the peak will not be reached until a time greater than $\tau$, or are you measuring delay relative to the peak of the impulse? – Floris Apr 4 '18 at 14:16